Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $k \neq 0$. $t = \dfrac{k - 1}{k + 7} \times \dfrac{k - 9}{k^2 - 10k + 9} $
Explanation: First factor the quadratic. $t = \dfrac{k - 1}{k + 7} \times \dfrac{k - 9}{(k - 1)(k - 9)} $ Then multiply the two numerators and multiply the two denominators. $t = \dfrac{ (k - 1) \times (k - 9) } { (k + 7) \times (k - 1)(k - 9) } $ $t = \dfrac{ (k - 1)(k - 9)}{ (k + 7)(k - 1)(k - 9)} $ Notice that $(k - 9)$ and $(k - 1)$ appear in both the numerator and denominator so we can cancel them. $t = \dfrac{ \cancel{(k - 1)}(k - 9)}{ (k + 7)\cancel{(k - 1)}(k - 9)} $ We are dividing by $k - 1$ , so $k - 1 \neq 0$ Therefore, $k \neq 1$ $t = \dfrac{ \cancel{(k - 1)}\cancel{(k - 9)}}{ (k + 7)\cancel{(k - 1)}\cancel{(k - 9)}} $ We are dividing by $k - 9$ , so $k - 9 \neq 0$ Therefore, $k \neq 9$ $t = \dfrac{1}{k + 7} ; \space k \neq 1 ; \space k \neq 9 $